Statistics and spin on two-dimensional surfaces

Abstract

The fundamental groups of the configuration spaces for the O(3) nonlinear σ-model on the compact genus g surfaces T^2g and on the connected sums R²#T^2g are known for any soliton number N. So are the braid for N spinless particles on these manifolds. The representations of these groups govern the possible statistics of solitons and particles. We show that when spin and creation/annihilation processes are introduced, the fundamental groups for the particles are the same as the corresponding σ-model groups. These fundamental groups incorporate the spin-statistics connection and are of greater physical relevance than the standard braid groups.